2. CHAPTER 5: Exponential and Logarithmic Functions
5.1 Inverse Functions
5.2 Exponential Functions and Graphs
5.3 Logarithmic Functions and Graphs
5.4 Properties of Logarithmic Functions
5.5 Solving Exponential and Logarithmic Equations
5.6 Applications and Models: Growth and Decay; and Compound Interest
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3. 5.4 Properties of Logarithmic Functions
Convert from logarithms of products, powers, and quotients to expressions in terms of individual logarithms, and conversely.
Simplify expressions of the type logaax and
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4. Logarithms of Products
The Product Rule
For any positive numbers M and N and any logarithmic base a,
loga MN = loga M + loga N.
(The logarithm of a product is the sum of the logarithms of the factors.)
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5. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Example
Express as a single logarithm:
Solution:
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6. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Logarithms of Powers
The Power Rule
For any positive number M, any logarithmic base a, and any real number p,
(The logarithm of a power of M is the exponent times the logarithm of M.)
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7. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Examples
Express as a product.
Solution:
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8. Logarithms of Quotients
The Quotient Rule
For any positive numbers M and N, and any logarithmic base a,
(The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.)
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9. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Examples
Express as a difference of logarithms:
Solution:
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10. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Examples
Express as a single logarithm:
Solution:
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11. Applying the Properties - Example
Express each of the following in terms of sums and differences of logarithms.
Solution:
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12. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Example (continued)
Solution:
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13. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Example (continued)
Solution:
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14. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Example
Express as a single logarithm:
Solution:
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15. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Examples
Given that loga 2 ≈ and loga 3 ≈ 0.477, find each of the following, if possible.
Solution:
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16. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Examples (continued)
Solution:
Cannot be found using these properties and the given information.
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17. Expressions of the Type loga ax
The Logarithm of a Base to a Power
For any base a and any real number x,
loga ax = x.
(The logarithm, base a, of a to a power is the power.)
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18. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Examples
Simplify.
a) loga a8 b) ln et c) log 103k
Solution:
a. loga a8
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19. Expressions of the Type
A Base to a Logarithmic Power
For any base a and any positive real number x,
(The number a raised to the power loga x is x.)
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20. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Examples
Simplify.
Solution:
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